# Constructing continuous-variable spacetime quantum states from measurement correlations. (arXiv:1903.06312v1 [quant-ph])

Quantum states, no matter pure states, density matrices or Wigner functions,

fully describe a physical system at a particular time. Time remains to be one

of the deepest mysteries in physics for centuries. Here we attempt to

understand time via a unified approach on both space and time, under the

intuition that relativity treats space and time on an equal footing; thus we

build quantum states across spacetime instead of only on spatial slices. We no

longer distinguish measurements on the same system at different times with

measurements on different systems at one time and construct spacetime states

upon these measurement statistics. As a first step towards quantum field

theory, we consider how to approach this in the continuous-variable multi-mode

regime. We propose six possible definitions for spacetime states in continuous

variables, based on four different measurement processes: quadratures,

displaced parity operators, position measurements and weak measurements. They

are motivated by the pseudo-density matrix formulation among indefinite causal

structures and the path integral formalism. We show that these definitions lead

to desirable properties, and suggest the differences and similarities between

spatial and temporal correlations. An experimental proposal for tomography is

suggested to demonstrate that the spacetime states we define have very strong

operational meanings. All these show that it is possible to define a

continuous-variable spacetime state with multiple desirable properties and with

the possibility to apply the arsenal of continuous-variable quantum state tools

to the analysis thereof.